To find the inverse of the function [tex]\( f(x) = 2x + 16 \)[/tex], we need to follow these steps:
### Step-by-Step Solution
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
First, write the equation [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = 2x + 16
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
To find the inverse, we need to solve this equation for [tex]\( x \)[/tex]. Start by isolating [tex]\( x \)[/tex] on one side:
[tex]\[
y = 2x + 16
\][/tex]
Subtract 16 from both sides:
[tex]\[
y - 16 = 2x
\][/tex]
Divide both sides by 2:
[tex]\[
x = \frac{y - 16}{2}
\][/tex]
3. Express the inverse function [tex]\( f^{-1}(x) \)[/tex]:
Now, replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the expression for [tex]\( x \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{x - 16}{2}
\][/tex]
### Final Expression
We are given that:
[tex]\[
f^{-1}(x) = \frac{x}{[?]} + []
\][/tex]
From our solution, we can see that:
[tex]\[
f^{-1}(x) = \frac{x - 16}{2}
\][/tex]
We can rewrite this expression in the form given in the problem:
[tex]\[
f^{-1}(x) = \frac{x}{2} - 8
\][/tex]
Thus, filling in the blanks, we obtain:
[tex]\[
f^{-1}(x) = \frac{x}{2} - 8
\][/tex]
So the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{x}{2} - 8
\][/tex]
